Nonlinear embeddings into Banach spaces

Nonlinear embeddings of Banach spaces has been an active field of research since the mid 20th century with many applications to theoretical computer science (Sparsest Cut problem, Nearest Neighbor Search etc), geometry (the Gromov’s positive scalar curvature conjecture, the Novikov conjecture etc) and group theory (growth of groups, amenability etc). In this dissertation, we review some pre-existing theory about isometric, bi-Lipschitz, quasi-isometric, and coarse embeddings of metric spaces into Banach spaces, as well as provide some new results. In Section 3 we calculate new optimal bounds from below for distortion fo $\ell_q$ into $p$-uniformly convex Banach spaces. In particular, this allows us to present a new proof of the fact that there exists a doubling subset of $\ell_q$ that does not admit any bi-Lipschitz embedding into $\bR^d$ for any $d\in\bN$ and $q>2$ (this fact follows from work by V. Lafforgue and A. Naor and independent results by Y. Bartal, L. Gottlieb, O. Neiman). We also study how our approach can be generalized to obtain embeddability obstructions into non-positively curved spaces. In Section 4 we study equivariant coarse embedding into $\ell_q$. Those are special kind of coarse embeddings which come with a representation that is connected to the embedding itself. We show that if a normed vector space, viewed as an abelian group under addition, admits an equivariant coarse embedding into $\ell_p$ then it also embeds in a bi-Lipschitz way into $\ell_p$. We discuss potential applications of this result to open problems about coarse embeddings into $\ell_p$